In particular, we simulate Planck Surveyor observations of a degree patch of sky at ten different observing frequencies. The sky emission includes contributions from primary CMBR fluctuations, kinetic and thermal Sunyaev--Zel'dovich effects from clusters and dust, free--free and synchrotron emission from the Galaxy.
Figure 19: The cluster profiles of some SZ effect reconstructions compared to the input profiles convolved with a 10 arcmin beam (solid line). The full MEM with full ICF information was used to reconstruct the dashed line whereas the quadratic approximation to this was used to reconstruct the dotted line.
The reconstructions presented in Section 3 show that, in the absence of severe point source contamination, the Planck Surveyor observations enable the recovery of the CMBR fluctuations with an absolute accuracy of about . Moreover, depending on assumed knowledge of the power spectra of the various components, we find that it is possible to reconstruct the emission due to other components with varying degrees of accuracy. In particular, the Galactic dust emission may be reconstructed with an accuracy of about . The main features of Galactic free--free are also reconstructed. We find that both the magnitude and radial profile of the thermal SZ effect may be recovered for rich clusters, but the reconstruction of the kinetic SZ effect is only possible in clusters which also have a large thermal SZ effect. Given the cluster gas profile derived from the thermal SZ effect, however, it may be possible to recover the kinetic effect more successfully by using optimal filtering methods tailored to individual cluster shapes (Haehnelt & Tegmark (1996)).
We find that in nearly all cases the MEM algorithm produces more accurate reconstructed maps and power spectra of the various components than the WF, and this is particularly true for reconstructions of the thermal SZ effect. This difference is most likely a result of the assumption in the WF method that the fields to be reconstructed are well-described by Gaussian random fields. This is clearly not the case for the SZ effect, and other foreground components such as Galactic dust also appear quite non-Gaussian in nature.
In creating the reconstructions presented in the is paper, we have throughout assumed that the frequency dependence of all the components are known a priori. This is reasonable for the CMBR emission, as well as the kinetic and thermal SZ effects, but it is unlikely to be the case for the three Galactic components. For real observations, the spectral indices of the free--free and synchrotron emission will be uncertain to within about 20 per cent. Moreover, the dust temperature and emissivity may be known to even poorer accuracy.
If we assume for the moment that the frequency dependence of each component is the same across the entire degree field, then we find that reasonable uncertainties in the parameters describing the Galactic components do not significantly affect our reconstructions. In fact, we find that the dust temperature and emissivity may be determined to within 1 per cent accuracy from the data by including them as free parameters in either the MEM or WF algorithm. The resulting reconstructions of all the physical components are virtually indistinguishable from those obtained by assuming these parameters. Unfortunately, we find that it is not possible to determine either the free--free or synchrotron spectral index in this way. Nevertheless, if in the algorithm we assume a spectral index for either component that is in error by within 20 per cent, we find that the reconstructions of the remaining components are virtually unaffected.
In this paper, the simulated Planck Surveyor observations were somewhat idealised in that it was assumed that the beam at each observing frequency was a simple Gaussian. For the real observations, however, it is unavoidable that the beam will infact possess sidelobes at some level, and care must be taken to incorporate any such features into the analysis, in particular if these sidelobes contain emission from any strong sources.
The simulated observations presented here also assume that any striping due to the scan strategy has been removed to a sufficient level so that it may be considered negligible. In fact, it may be possible to use MEM to perform the destriping of the maps and the component separation simultaneously. Indeed the simultaneous reconstruction of a deconvolved CMBR maps and the removal of scan baselines has already been performed using MEM in the analysis of Tenerife data (Jones et al. (1998)).
In terms of computational speed, however, the most important assumption made in our simulations were that the beam at each frequency does not change shape with position on the sky This assumption is not unreasonable in the analysis of small patches of sky considered here, but may be questionable for all-sky maps. The importance of this assumption lies in the fact that the beam-smoothing may be written as a convolution and therefore allows us to analyse the observations entirely in the Fourier domain, where each mode may be considered independently. As discussed in the companion paper, this means that the analysis is reduced to a large number () of small-scale linear inversion problems and so is computationally very fast.
If the beam is spatially varying, however, the beam-smoothing cannot be written as a simple convolution. In this case the analysis should properly be performed in the sky plane, and requires the use of sparse matrix techniques to compute the beam-smoothing at each point on the sky as opposed to Fast Fourier Transforms (FFTs). In addition, the matrices involved in the linear problem are then very large indeed, since we are attempting simultaneously to determine parameters by the minimisation of a function of corresponding dimensionality. Nevertheless, the authors have investigated the use of MEM and the WF in this case and find that reconstructions similar to those presented here can be performed in about 12 hours of CPU on a SPARC 20 workstation, For both MEM and WF, however, the calculation of errors cannot be performed by inverting the Hessian matrix of the posterior probability, since this matrix has dimensions . Although this matrix is in fact reasonable sparse, the inversion is still not feasible. Instead, the errors on the reconstructions must be estimated by performing several hundred Monte-Carlo simulations for different noise realisations (see Maisinger et al. (1997)).
Finally, perhaps the most important improvement on the simulations and reconstructions presented here is the inclusion of a realistic population of point sources. Using the point source simulations of Toffolatti et al. (1998), a full investigation of the effects on the reconstruction of the CMBR and other components will be presented in a forthcoming paper (Hobson et al, in preparation).